Congruence Primes of the Kim–Ramakrishnan–Shahidi Lift

ABSTRACT

For a primitive form f of weight k for let KS(f) be the Kim–Ramakrishnan–Shahidi (K–R–S) lift of f to the space of cusp forms of weight for Based on some working hypothesis, we propose a conjecture, which relates the ratio of the periods (Petersson norms) to the symmetric 6th L-value L(3k − 2, f, Sym⁶) of f. From this, we also propose that a prime ideal dividing the (conjectural) algebraic part L(3k − 2, f, Sym⁶) of L(3k − 2, f, Sym⁶) gives a congruence between the K–R–S lift and non-K–R–S lift, and test this conjecture numerically.

2000 AMS Subject Classification:

• Primary 11F67
• 11F46
• 11F66

Keywords:

• Kim-Ramakrishnan-Shahidi lif
• symmetric 6-th L function
• congruence

Acknowledgments

The authors thank the referee for many valuable comments.

Funding

The first author was partially supported by JSPS KAKENHI Grant Numbers 24540005 and 25247001, and both authors were partially supported by JSPS KAKENHI Grant Number 23224001.

Notes

¹ This was first informed by A. Mellit.